EXIT-constrained BICM-ID design using extended mapping
© Fukawa et al; licensee Springer. 2012
Received: 28 April 2011
Accepted: 9 February 2012
Published: 9 February 2012
This article proposes a novel design framework, EXIT-constrained binary switching algorithm (EBSA), for achieving near Shannon limit performance with single parity check and irregular repetition coded bit interleaved coded modulation and iterative detection with extended mapping (SI-BICM-ID-EM). EBSA is composed of node degree allocation optimization using linear programming (LP) and labeling optimization based on adaptive binary switching algorithm jointly. This technique achieves exact matching between the Demapper (Dem) and decoder's extrinsic information transfer (EXIT) curves while the convergence tunnel opens until the desired mutual information (MI) point. Moreover, this article proposes a combined use of SI-BICM-ID-EM with Doped-ACCumulator (D-ACC) and modulation doping (MD) to further improve the performance. In fact, the use of D-ACC and SI-BICM-ID (noted as DSI-BICM-ID-EM) enables the right-most point of the EXIT curve of the combined demapper and D-ACC decoder (Ddacc), denoted as DemDdacc, to reach a point very close to the (1.0, 1.0) MI point. Furthermore, MD provides us with additional degree-of-freedom in "bending" the shape of the demapper EXIT curve by choosing the mixing ratio of modulation formats, and hence the left most point of the demapper EXIT curve can flexibly be lifted up/pushed down with MD aided DSI-BICM-ID-EM (referred to as MDSI-BICM-ID-EM). Results of the simulations show that near-Shannon limit performance can be achieved with the proposed technique; with a parameter set obtained by EBSA for MDSI-BICM-ID-EM, the threshold signal-to-noise power ratio (SNR) is only roughly 0.5 dB away from the Shannon limit, for which the required computational complexity per iteration is at the same order as a Turbo code with only memory-2 convolutional constituent codes.
The discovery of Turbo code  in 1993 is a landmark event in the history of coding theory, since the code can achieve near-Shannon limit performance. It is shown in  that the Turbo code composed of memory-4 constituent convo-lutional codes can achieve 0.7 dB, in Signal-to-noise power ratio (SNR), away from the Shannon limit. Various efforts have been made since then to achieve Turbo code-like performance without requiring heavy computational efforts for decoding.
Bit-interleaved coded modulation and iterative detection/decoding (BICM-ID)  has been recognized as a bandwidth efficient coded modulation scheme, of which transmitter is comprised of a concatenation of encoder and bit-to-symbol mapper separated by a bit interleaver. Iterative detection-and-decoding takes place at the receiver, where extrinsic log likelihood ratio (LLR), obtained as the result of the maximum a posteriori probability (MAP) algorithm for demapping/decoding, is forwarded to the decoder/demapper via de-interleaver/interleaver and used as the a priori LLR for decoding/demapping according to the standard turbo principle.
Performances of BICM-ID have to be evaluated by the convergence and asymptotic properties , which are represented by the threshold SNR and bit error rate (BER) floor, respectively. In principle, since BICM-ID is a serially concatenated system, analyzing its performances can rely on the area property  of the EXtrinsic Information Transfer (EXIT) chart. Therefore, the transmission link design based on BICM-ID falls into the issue of matching between the demapper and decoder EXIT curves.
Various efforts have been made seeking for better matching between the two curves to minimize the gap, while still keeping the tunnel open, aiming, without requiring heavy detection/decoding complexity, at achieving lower threshold SNR and BER floor. In , ten Brink et al. introduced a technique that makes good matching between the detector and decoder EXIT curves using low density parity check (LDPC) code in multiple input multiple output (MIMO) spatial multiplexing systems.
It has long been believed that for 4-quadrature amplitude modulation (4-QAM), the combination of Gray mapping and Turbo or LDPC codes achieves the optimal performance. However, Schreckenbach et al.  propose another approach towards achieving good matching between the two curves by introducing different mapping rules, such as non-Gray mapping, which allows the use of even simpler codes to achieve BER pinch-off (corresponding to the threshold SNR) at an SNR value relatively close to the Shannon limit.
Another technique that can provide us with the design flexibility is extended mapping (EM) presented in [7, 8] where with 2 m -QAM, ℓmap bits (ℓmap > m), are allocated to one signal point in the constellation. With EM, the left-most point of the demapper EXIT function has a lower value than that with the Gray mapping, but the right-most point becomes higher. With this setting, the demapper EXIT function achieves better matching even with weaker codes such as short memory convolution codes as shown in . However, there is a fundamental drawback with the structure shown in ; it still suffers from the BER floor simply because the demapper EXIT curve does not reach the top-right (1.0,1.0) MI point.
In , Pfletschinger and Sanzi suggest that by using the memory-1 rate-1 recursive systematic convolutional code (RSCC), referred to as D-ACC located immediately after the interleaver, the error floor can be eliminated. Furthermore, it was shown by  that by replacing the RSCC-coded bits b u (P) with the accumulated bits b c (P) at every P bit-timings, the technique of which is referred to as inner doping with doping ratio (1:P), the EXIT curve of DemDdacc can be flexibly changed.
Several techniques have been proposed to determine optimal labeling pattern for the modulation (bit pattern vector allocated to each constellation point). The ideas of binary switching algorithm (BSA), which aims at labeling costs optimization, are presented in [6, 11]. However, the BSA based labeling optimization evaluates the labeling cost assuming that full a priori information is available. Hence, this approach only aims at lifting up as much the rightmost point of the demapper EXIT curve as possible. Yang et al.  introduce adaptive binary switching algorithm (ABSA) to obtain optimal labeling pattern, where optimality is defined by taking into account the labeling costs at multiple a priori MI points. Hence, ABSA changes the shape of the demapper EXIT curves more flexibly than BSA. However, the optimal labeling obtained in ABSA is on given code-basis since the code parameter optimization is not included in the ABSA iterations.
In our previous publication , we introduced a BICM-ID technique that uses even simpler codes, single parity check code (SPC) and irregular repetition code (IRC), combined with EM. For the notation simplicity, we refer our proposed BICM-ID structure in  to as SPC-and-IRC aided BICM-ID with EM (SI-BICM-ID-EM). We investigated in  that linear programming (LP) technique can be applied for SI-BICM-ID-EM to determine the optimal degree allocations for the IRC code with the aim of achieving desired convergence property. Moreover, in  we proposed a combined use of modulation doping (MD), originally proposed in [16, 17], which mixes the labeling rules for the extended non-Gray mapping and the standard Gray mapping at a certain ratio. The technique proposed in  helps the left-most point of the demapper slightly be lifted up to initiate the LLR exchange between the demapper and the decoder. This technique gives the additional degree-of-freedom in "bending" the shape of the demapper EXIT curve by choosing the mixing ratio and hence the left-most point of the demapper EXIT curve can be flexibly lifted up/pushed down. This article proposes a combined use of SI-BICM-ID-EM with D-ACC and MD. The D-ACC aided SI-BICM-ID-EM is referred to as DSI-BICM-ID-EM, and MD aided DSI-BICM-ID-EM is referred to as MDSI-BICM-ID-EM later on.
The primary goal of this article is to create a design framework for the optimization of SI-BICM-ID-EMa by combining those techniques into a unified iterative algorithm. To achieve the goal described above, this article proposes a new labeling pattern optimization technique, EXIT-constrained Binary Switching Algorithm (EBSA). The gap between the two EXIT curves is taken into account in a repeat-until loop that controls the EBSA algorithm. Hence, the process for determining the optimal degree allocation using LP [13, 14] is also included in the repeat-until loop in EBSA.
The results of simulations show that near-Shannon limit performance can be achieved with the proposed techniques; BER simulation results show that 4-QAM EM with ℓmap = 5, the threshold SNR is only roughly 0.5 dB away from the Shannon limit with MDSI-BICM-ID-EM, for which required computational complexity for DemDdacc is almost the same as a Turbo code with only memory-2 convolutional constituency codes, per iteration.
This article is organized as follows; our proposed system structure is described in Section 2. Theoretical EXIT functions of the codes used in SI-BICM-ID-EM are presented in Section 3. EBSA is introduced and detailed in Section 4, which is the core part of this contribution. In Section 5 numerical results are provided: in Section 5.1, convergence property of the proposed schemes described to confirm the effectiveness of EBSA; in Section 5.2, the results of BER performance evaluations are presented. In Section 6, computational complexity with the proposed technique is assessed briefly. Finally, we conclude this article in Section 7 with some concluding statements.
2 System model
The repetition times d v , referred to as variable node degree, may take different values in a block (transmission frame); if d v takes several different values in a block, such code is referred to as having irregular degree allocations. It is assumed throughout this article that d c takes only one identical value as in .
bits per symbol, where a i represents the ratio of variable nodes having degree in a block and M is the number of node degree allocations.
The coded bit sequence is bit-interleaved, and segmented into ℓmap-bit segments, and then each segment is mapped on to one of the 2 m constellation points for modulation. The complex-valued signal modulated according to the mapping rule is finally transmitted to the wireless channel. Since ℓmap > m with EM, more than one label having different bit patterns in the segment are mapped on to each constellation point. However, there are many possible combinations of the bit patterns, hence determining of the optimal labeling pattern plays the key role to achieve limit-approaching performance.
where is labeling pattern and ψ(·) is the mapping function as indicated in Figure 1. n is zero mean complex AWGN component with variance (i.e.,〈|x(k)|2〉 = 1, 〈n(k)〉 = 0 and ) for ∀k.
where S0 (S1) indicates the set of the labeling pattern s having the v th bit being 0(1), and La,Dem(b ρ (s)) is the demapper's a priori LLR fed back from the decoder corresponding to the ρ th position in the labeling pattern s.
Decoding takes place segment-wise where, because of the irregular code structure, the variable node degrees have different values segment-by-segment. Structure of the decoder as well as decoding algorithm is detailed in the previous publications, e.g., in [13, 14, 20]. Therefore, only summary of the algorithm is provided in this article.
This process is performed for the other variable nodes in the same segment having the variable node degree , and also for all the other segments independently in the same transmission block. Finally, the updated extrinsic LLRs obtained at the each variable node are interleaved, and fed back to the demapper. For the final bit-wise decision, a posteriori LLR output from the decoder is used.
Reference  proposes a combined use of D-ACC with SI-BICM-ID-EM (DSI-BICM-ID-EM).b The purpose of introducing D-ACC is to lift the right-most point of demapper EXIT curve up to reach the (1.0,1.0) MI point so that the BER floor with SI-BICM-ID-EM can be eliminated. In this system structure, D-ACC is placed between the interleaver and mapper as shown in Figure 1 of . The coded bit sequence is bit-interleaved, and input to the D-ACC with doping ratio of (1:P). To keep the D-ACC's code rate equal to one, the interleaver's output is replaced by a D-ACC-coded bit at every P th bit.
3 EXIT analysis
Since detailed investigation for the effect of EM on the shape and the demap-per's EXIT function is provided in the previous publications, e.g., , they are not provided in this article. For those readers who are interested in this issue can refer [7, 13, 20].
Hence, it is found that the key of achieving the best matching between the demapper and the decoder EXIT curves is to jointly optimize the labeling pattern and the variable node degree distribution a i .
4 Framework for EBSA-based DSI-BICM-ID-EM design
4.1 Optimal node degree allocation using LP
More details are given in Appendix 1. Furthermore, to find the optimal check node degree d c , this article proposes a brute-force search (all possible value search),c as summarized in Algorithm 1.
Algorithm 1 Optimal degree allocation algorithm
Initialize and a i values.
for d c = 2 to max d c do
Perform LP for Equation (14) with fixed d c and obtain optimal distribution a i for each .
Calculate code rate R using d c , d v and a i .
Find and aopt achieving R → max.
return and aopt.
4.2 EBSA framework
In , Yang et al. introduce the idea of Adaptive BSA (ABSA) which takes into account the costs at multiple a priori information points. The gap width between the demapper and the decoder EXIT curves is also taken into account, given the decoder EXIT curve. ABSA then obtains the optimal doping ratio in conjunction with determining the optimal labeling pattern. Hence, opening of the convergence tunnel until the (1.0, 1.0) MI point is guaranteed with this technique. However, ABSA does not change the code parameters in optimization process, and therefore, optimality is only on given code-basis.
Since both the ABSA and EBSA algorithms, in common, are based on the BSA, as well as the same cost definition, BSA and the cost are summarized in Appendices 2 and 3, respectively, for the completeness of the article. This article's proposed EBSA algorithm is summarized in Algorithm 2. It should be noticed that the processes for determining the optimal doping ratio, the d c value, and the LP based code parameter optimization are all included in a single repeat-until loop. This indicates that the code parameters are also changed in the EBSA framework.
It should be further noticed that the horizontal and vertical gap widths evaluation, as descriptively summarized in Figure 5, is included in the repeat-until loop. With the EBSA framework, the labeling pattern used in the LP-based degree allocation optimization for DSI-BICM-ID-EM are obtained by lowering the cost of (at right-most MI point corresponding to the case with full a priori information) as much as possible, while still keeping the vertical gap smaller than the predefined value δ w . Hence, other costs are ignored in the LP based optimization.
5 Numerical results
5.1 Convergence property analysis
The EBSA optimization technique is a design framework for BICM-ID, and therefore applicable not only to MDSI-BICM-ID-EM, but also to other structures, as described in Endnote "a" in Section 1. To demonstrate the performances superiority with the optimization techniques described in this article, EXIT curves were calculated for several designs described in the previous sections, aiming at the turbo cliff to happen at SNR = 0.8 dB and 3.1 dB, as examples.
5.1.1 SI-BICM-ID-EM with node degree optimization using LP
Initial degree distributions
Optimized degree distributions
Algorithm 2 EXIT-constrained binary switching algorithm (EBSA)
Initialization: Generate labeling pattern s randomly as well as degree distribution d v empirically
for P = 2 and d c = 2 to max P and max d c , respectively do
Draw demapper EXIT curve based on the given labeling pattern s obtained as the result of the latest iteration
for i = 1 to Nmaxd do
Conduct BSA with and
Draw the demapper's EXIT curve using the obtained labeling pattern.
Draw the decoder's EXIT curve using degree distribution using LP for Equation (14) with the obtained labeling pattern.
λ q = λ q -- 1
Select the labeling pattern and decoder node distribution that has minimum gap.
until Gap between the demapper and decoder EXIT curve becomes smaller than the threshold for each MI points tested.
Select the optimal parameter set that minimizes the gap
It is found by carefully looking at the right-most part of the curves that the intersection point of the (ii)-(iv) decoder curves and the demapper curve indicated by (*) in the figure are found to be slightly closer to the extrinsic MI = 1.0 than the empirically designed case. However, the rate of the code obtained by LP is slightly lower than the rate of the code with empirically obtained degree allocation, and the intersection point of the two EXIT curves is still quite apart from the (1.0,1.0) MI point. Therefore, LP alone can lower the BER floor, but cannot increase the spectrum efficiency in those cases.
5.1.2 DSI-BICM-ID-EM with node degree optimization using LP technique
Similar result can be observed from Figure 8b, where optimization was performed for SNR = 3.1 dB. Both in Figures 8a,b, the two EXIT curves intersect at a point very close to the (1.0,1.0) MI point. Therefore, no BER floor (or, at least invisible in the BER value range shown in the figure) and higher spectrum efficiency compared to the empirically designed SI-BICM-ID-EM are expected.
5.1.3 MDSI-BICM-ID-EM with EBSA
From Figure 10a, very close matching between the DemDdacc and the decoder EXIT curves can be observed from the starting point to the end. Moreover, now the DemDdacc EXIT curve starts from (0,0.0064) and thereby, LLR exchange can be initiated, and hence the trajectory can reach the target MI point close enough to the (1.0,1.0) point. Similar characteristics can be observed in Figure 10b, where the optimization was performed for SNR = 3.1 dB.
5.2 BER performances
6 Complexity assessment
With ℓmap = 5, there are 32 labeling patterns in total, where each of the sets S0 and S1 contains 16 patterns. Hence, the probabilities for the 16 patterns have to be summed up when calculating the numerator and the denominator of (5). Since the BCJR algorithm requires forward and backward processing and each state emits two branches corresponding to the systematic input being 0 and 1, the computational complexity for the demapper having 16 labeling patterns both in the numerator and the denominator is equivalent to the decoding complexity of memory-3 convolutional code using the BCJR algorithm . Furthermore, since Turbo code requires at least two constituency codes , the complexity estimated above is also roughly equivalent to that required by a Turbo code having two memory-2 constituent convolutional codes .
Calculation cost of 4-QAM EM
This article has proposed a design framework, EBSA, and applied it to our proposed BICM-ID techniques, SI-BICM-ID-EM, DSI-BICM-ID-EM and MDSI-BICM-ID-EM. Since EBSA takes into account the horizontal and vertical gap widths between the DemDdacc and decoder's EXIT curves at the pre-defined several MI points, it can determine the optimal labeling pattern for EM and degree allocations simultaneously. In fact, when EBSA is applied to DSI-BICM-ID-EM, two curves exactly match, and surprisingly the left-most point of the EXIT curve of DemDdacc is determined to be the (0.0, 0.0) MI point, and hence LLR exchange can not be initiated. To avoid this situation, this article introduced the MD technique, by which the left-most point of the DemDdacc EXIT curves can be lifted up slightly while the other part still exactly matched. As the result, very close-Shannon limit performance can be achieved without requiring heavy computational burden with ℓmap = 5 EM 4-QAM; the complexity is almost the same level as a Turbo code with only memory-2 constituency codes.
The following three issues have to be noted in concluding this article, since this special issue has two focal points, "Algorithm and Implementation Aspects":
The proposed EBSA is applicable to BICM-ID techniques using other codes, so far as the degree allocation optimization can be performed using LP. LDPC-aided BICM-ID  and irregular convolutional code-aided BICM-ID  belong to this category. This is the reason why call EBSA "framework" rather than "technique".
The trade-off between performance and complexity due to iterations can well be managed with EBSA by properly setting the horizontal and vertical gap parameters, ϵ and δ, respectively, at several MI points. Even relatively large gap parameters are used so that not too many iterations are required, still arbitrary low BER can be achieved because the two curves reach a point close enough to the (1.0, 1.0) MI point.
Application of the EBSA framework to higher order modulation is left as future study.
Appendix 1: Node degree optimization using LP
Code rate has to be lower than but as close to the capacity as possible.
Keep the convergence tunnel open between the demapper and decoder EXIT curve until the desired intersection point and the point should be as close to the (1.0, 1.0) MI point as possible.
Total of node degrees distributions has to be always 1.
Now, given the fact that the optimization parameter in (19) is only a i and the other terms are constant and furthermore, the index and constraints are both linear function of the optimization variable a i . Hence, the problem can be solved by using LP techniques.
Appendix 2: Summary of BSA
for . The BSA is summarized in Algorithm 3.
Algorithm 3 Binary switching algorithm (BSA)
Initialization: generate labeling pattern randomly.
Select the symbol s high which has the highest cost .
Find the symbol s low which can achieve maximum reduction of the total
cost by swapping the positions of s high and s low .
if s low exists. then
Swap s high and s low .
Update according to the new labeling pattern.
Set the symbol with the second highest cost as s high .
until There is no pair of symbols to switch
A problem with this approach is, however, that the cost is calculated assuming the availability of a full a priori information and thus, affect only the right most point of the DemDdacc EXIT curve, and it does not consider the matching between the demapper and decoder EXIT curves. This approach is reasonable only when the objective is to force the right-most point of the demapper curve to reach as close to the (1.0,1.0) MI point as possible. However, this article has already proposed the use of D-ACC which already makes it possible for the demapper EXIT curve to reach a point close enough to the (1.0,1.0) MI point. In the subsection 4-D, a novel algorithm, EBSA, is introduced to obtain a labeling pattern aiming at better matching between the two curves, assuming the use of D-ACC.
Appendix 3: Cost functions for ABSA and EBSA
Those costs affect the shape of the demapper EXIT curve. Figure 6 shows an intuitive example for ℓmap = 5.
aThe proposed technique is also applicable to other BICM-ID techniques, so far as they use LP for degree allocation optimization. LDPC aided BICM-ID as well as irregular convolutional code aided BICM-ID belong to this class. This is the reason on why we call the technique proposed in this article "framework". bCombined use of D-ACC with BICM-ID itself was first proposed by . The technique is also used in  to determine the optimal doping rate when it is combined with BSA. c d c has indirect effect to the LP optimization, but to the code rate. dSchreckenbach et al.  show that Nmax = 100 is enough. eThe range is defined as: MI(Z q + ΔZq) for q = 0, MI(Z q ± ΔZ q ) for 1 ≤ q ≤ ℓmap -- 2 and MI(Z q -- ΔZ q ) for q = ℓmap -- 1, where Δ = 1/(ℓmap -- 1).
This research was in part supported by the Japanese government funding program, Grant-in-Aid for Scientific Research (B) No. 20360168 and (C) No. 22560367, and in part by Finland distinguished professor program funded by Finnish National Technology Agency Tekes. The authors are highly thankful for valuable technical comments and suggestions given by Mr. Takehiko Kobayashi of Hitachi Kokusai Electric Inc. We also acknowledge Mr. Xin He of Information Theory and Signal Processing Lab., School of Information Science, JAIST for his valuable opinions and suggestions to improve the quality of this article.
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